On a study on dynamics of localized pattern for a bistable reaction diffusion equations with a heterogeneous environment

本文研究非线性扩散方程的自由边值问题。这些问题可以用来描述生物或化学物种的扩散，自由边界代表扩展的前沿。如果非线性是单稳态的、非线性的或燃烧型的，杜教授和楼教授对问题的长时间动力学行为得到了比较完整的描述，并揭示了所谓的“扩散”和“消失”之间的急剧过渡现象。利用“半波”的概念确定了传播时波前的渐近传播速度。在这项研究中，我们给出了一个更尖锐的估计的前沿的扩展速度比在杜和娄的工作，我们描述了如何解决方案接近半波时，发生扩展。对于（1）一维问题，（2）具有径向对称设置的高维问题和（3）一维对流扩散问题，我得到了这些结果。

In this research, we study free boundary problems for nonlinear diffusions equations. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundaries representing the expanding fronts. If the nonlinearity is monostable, bistable, or combustion type, Professor Du and Professor Lou obtained a rather complete description of the long-time dynamical behavior of the problem and revealed sharp transition phenomena between so called "spreading" and "vanishing". They also determined the asymptotic spreading speed of the fronts by using of "semi-waves" when spreading happens. In this research we we give a much sharper estimate for the spreading speed of the fronts than that in the work of Du and Lou, and we describe how the solution approaches the semi-wave when spreading happens. I obtained these results for (1)1 dimensional problem, (2)higher dimensional problem with radially symmetric setting and (3)1 dimensional advection-diffusion problem.